Friday, 4 July 2014

An arbitrary quantum cannot be cloned

One of the early important results in the study of quantum information is the no-cloning theorem, which tells us that there is no quantum operation that allows us to create multiple copies of an arbitrary quantum state.

This property is very different from what we expect from classical information, which you may reproduce as many times as you wish. For example, you can send a PDF file by email to many recipients while keeping a copy to yourself. The important point is that whatever the contents may be, you can make a duplicate of it.

Now consider a cloning machine M for qubits that can produce identical copies of the states |u) and |d):

M |u) |0) = M |u) |u),
M |d) |0) = M |d) |d) ,

where |0) denotes any fixed initial state for M. This is necessary in pretty much the same way you would need a blank piece of paper before you can photocopy a printed document.

Friday, 13 June 2014

How to teleport a qubit

One of the fascinating things we can do with quantum entanglement is a scheme called quantum teleportation. In the original proposal by Charlie Bennett, Gilles Brassar, Claude Crepeau, Richard Jozsa, Asher Peres and Bill Wootters, it describes a way to transmit an arbitrary quantum state between two parties who may be far apart, using only a Bell state shared between the two parties, a few qubit operations that each party can perform independently, and two bits of information that can be communicated by one party to the other.

Suppose Alice and Bob are in separate locations but they share a pair of electrons that are in the entangled state

|E) = (|u,u) + |d,d)) / sqrt(2)

where as usual |u) denotes the state of an electron having its spin pointing in the up-direction, |d) denotes that with spin in the down-direction, and |u,u) refer to the state of the first and second electrons, respectively. Let's say that Bob has the first electron on his side and Alice has the second electron on her side. 

Alice also possesses a third electron in the state

|q) = a |u) + b |d)

and she wants Bob to obtain this state. If Alice does not know what the value of a and b precisely, she can not clone the state and send a copy to Bob. However, since Alice and Bob have shared entanglement, it is possible to transfer the state of this electron into Bob's electron using teleportation, which is shown in the figure below. 

Thursday, 13 March 2014

A quantum version of Zeno's paradox

The quantum Zeno effect describes the situation where an unstable particle, say a radioactive atom, won’t decay if it is observed continuously. More generally, it says that if you repeatedly interact with a quantum system through measurement then you can effectively freeze its quantum state.

The phenomenon is named after a Greek philosopher of ancient times, Zeno of Elea. Zeno is known best for a set of paradoxes (we know of 9 of them) that he posed as arguments  against Aristotle’s concept of motion. Here we are interested in the arrow or fletcher’s paradox.

If you observe an arrow flying through the air at some particular instant in time, then it would have a definite position, meaning it isn’t moving at that specific moment. However, you can think of the arrow’s motion as happening one moment at a time. This says that motion must be impossible since it is made up of this long sequence of motionless moments. 

Of course, as far as we can tell, the world is not static and objects in it are not forever motionless. What’s lacking with Zeno’s assertion is the mathematical notion of continuity. Motion is possible because time doesn’t flow like a series of separate frames in a film but more like the seamless current of a steady stream.

Saturday, 8 March 2014

Spins, magnets, and quantum mechanics

Quantum mechanics is often described as an area of physics that deal with energy and matter at the atomic scales, where different weird, unusual stuff happen. To some extent, it is true that quantum objects behave in ways that seem counter to our everyday, common-sense intuition. Unfortunately, focusing on these particular aspects of quantum theory might give the impression that it is mysterious, mystical, and difficult to understand. And that is simply not true. Things like superposition require a little getting used but, for the most part, quantum mechanics works in ways you expect and that naturally make sense. 

I hope to demonstrate this by discussing an experiment with atoms and magnets that is explained in fairly simple terms using quantum mechanics.  This is largely how I was  introduced to the subject many years ago. But to start, we will need to go over some basic ideas regarding magnets and magnetic fields.

Most of us are familiar with magnets from their ability to attract iron and similar metals but in physics, a magnet is just any material that produces a magnetic field. A magnet influences its surroundings through the magnetic field it creates and reacts to the magnetic fields it experiences from other magnetic objects.

A nice thing about magnets is we can understand how they work without having to be very precise. I'm sure many of you have played around with a bar magnet before, which is often found bent into a horseshoe shape, to create a region of particularly strong magnetic field in between the ends labeled north pole and south pole. You probably also know about and maybe experienced first-hand how opposite poles attract and similar ones repel, and how they attract or repel more when two magnets are brought closer together.  
 
Something less familiar is what determines the force at which a magnet attracts and repels objects. The strength of a magnet is measured by a property called magnetic moment, which is responsible for a magnet's tendency to align with magnetic fields.

Monday, 3 March 2014

Orthogonal states and quantum contextuality

In this post, we use the idea of orthogonal quantum states to describe a fascinating result in quantum mechanics known as the Kochen-Specker theorem. To begin, we review a bit of necessary mathematics.

Recall our usual example of a qubit represented by the spin of an electron. Typically, write the state |E) of such an electron as a superposition of the spin pointing up, which we write as the state |u), and spin pointing down, which we write as the state |d):

|E) = a |u) + b |d),

where a and b are numbers such that |a|^2 is the probability of measuring the spin as up and |b|^2 is the probability of measuring the spin as down.

We've seen before that the set of all possible states for a qubit corresponds to all possible directions the spin can point to, which can be described by using points on a sphere. However, we may choose to write our qubit states using any pair of polar opposite points, and normally we would choose the directions corresponding to north and south pole, which are labeled as the spin states |u) and |d), respectively.

Thursday, 27 February 2014

That remarkable theorem by Bell

Before quantum physics came along, assigning a state to a physical system meant that you can uniquely determine the value of any measurable property using that state. In quantum mechanics, however, we've learned this is not the case: even if you know the quantum state exactly, for most measurements the best you can hope for is to estimate the probabilities for the various possible outcomes.

You might wonder, is the inability to uniquely determine measurement outcomes just a lack of complete knowledge about the system? Or put differently, could you gain some additional information beyond the quantum state that will help you determine outcomes exactly? The answer is "no", quantum phenomena are fundamentally probabilistic in nature and one way to show this is using an argument first made by John Bell in the 1960s.

Bell's theorem, as the result is called, says that there is no hidden mechanism that decides the outcome of a quantum measurement. The way the argument works is that if you assume that there is some  unknown factor that manufactures the statistical results of a measurement, then you can establish a limit on the amount of correlation among certain properties you can measure, which you can state as a Bell inequality. You can then show that there are always some quantum states that can violate the inequality. In this post, we attempt to describe a simple example of Bell's theorem at work.


Monday, 24 February 2014

Understanding quantum entanglement

As mentioned before, a quantum computer exploits the rules of quantum theory for accomplishing feats that are considered impossible with conventional computers. Two features of quantum mechanics are often involved in achieving such feats: superposition, which was discussed in our last post, and entanglement, which is the topic here.

Quantum entanglement is considered to be one of the counterintuitive aspects of quantum mechanics. However it is not a particularly difficult concept to grasp if we start with the notion of correlation. Roughly speaking, entangled quantum systems are objects whose properties so strongly correlated that using a state to describe all of them as a single unit describes them better than assigning a state to individual parts.